Five letters, none in its own envelope

Five letters and their five matching addressed envelopes get mixed up, and the letters are stuffed into the envelopes completely at random, one per envelope. Find the probability that no letter is in its correct envelope.

$11/30$. This counts derangements: by inclusion–exclusion the number of permutations of $5$ with no fixed point is $5!\left(1-1+\tfrac1{2!}-\tfrac1{3!}+\tfrac1{4!}-\tfrac1{5!}\right)=44$, and $\tfrac{44}{120}=\tfrac{11}{30}$.